Properties

Label 378.2.g
Level 378
Weight 2
Character orbit g
Rep. character \(\chi_{378}(109,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 20
Newform subspaces 8
Sturm bound 144
Trace bound 5

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Defining parameters

Level: \( N \) = \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 378.g (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 8 \)
Sturm bound: \(144\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(378, [\chi])\).

Total New Old
Modular forms 168 20 148
Cusp forms 120 20 100
Eisenstein series 48 0 48

Trace form

\( 20q - 10q^{4} - 14q^{7} + O(q^{10}) \) \( 20q - 10q^{4} - 14q^{7} + 2q^{10} + 12q^{13} - 10q^{16} + 16q^{19} + 4q^{22} - 8q^{25} + 4q^{28} - 12q^{31} - 8q^{34} - 14q^{37} + 2q^{40} + 100q^{43} - 8q^{46} + 38q^{49} - 6q^{52} - 104q^{55} - 16q^{58} - 18q^{61} + 20q^{64} - 10q^{67} - 50q^{70} - 16q^{73} - 32q^{76} - 18q^{79} + 40q^{85} - 2q^{88} + 60q^{91} - 24q^{94} + 32q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(378, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
378.2.g.a \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-3\) \(-4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
378.2.g.b \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-4\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
378.2.g.c \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(1\) \(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
378.2.g.d \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-2\) \(1\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots\)
378.2.g.e \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(-4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
378.2.g.f \(2\) \(3.018\) \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(3\) \(-4\) \(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
378.2.g.g \(4\) \(3.018\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(-2\) \(0\) \(2\) \(0\) \(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(1-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
378.2.g.h \(4\) \(3.018\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(2\) \(0\) \(-2\) \(0\) \(q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(378, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(378, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( ( 1 - T + T^{2} )^{2} \))
$3$ 1
$5$ (\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \))(\( 1 - 2 T + 12 T^{3} - 29 T^{4} + 60 T^{5} - 250 T^{7} + 625 T^{8} \))(\( 1 + 2 T - 12 T^{3} - 29 T^{4} - 60 T^{5} + 250 T^{7} + 625 T^{8} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))
$11$ (\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 2 T - 12 T^{2} - 12 T^{3} + 91 T^{4} - 132 T^{5} - 1452 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 - 2 T - 12 T^{2} + 12 T^{3} + 91 T^{4} + 132 T^{5} - 1452 T^{6} - 2662 T^{7} + 14641 T^{8} \))
$13$ (\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} + 52 T^{3} + 169 T^{4} )^{2} \))
$17$ (\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 2 T - 24 T^{2} - 12 T^{3} + 427 T^{4} - 204 T^{5} - 6936 T^{6} + 9826 T^{7} + 83521 T^{8} \))(\( 1 - 2 T - 24 T^{2} + 12 T^{3} + 427 T^{4} + 204 T^{5} - 6936 T^{6} - 9826 T^{7} + 83521 T^{8} \))
$19$ (\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( ( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \))(\( ( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 8 T + 30 T^{2} - 96 T^{3} - 845 T^{4} - 2208 T^{5} + 15870 T^{6} + 97336 T^{7} + 279841 T^{8} \))(\( 1 - 8 T + 30 T^{2} + 96 T^{3} - 845 T^{4} + 2208 T^{5} + 15870 T^{6} - 97336 T^{7} + 279841 T^{8} \))
$29$ (\( ( 1 + 3 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 7 T + 29 T^{2} )^{2} \))(\( ( 1 + 7 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 3 T + 29 T^{2} )^{2} \))(\( ( 1 - 10 T + 76 T^{2} - 290 T^{3} + 841 T^{4} )^{2} \))(\( ( 1 + 10 T + 76 T^{2} + 290 T^{3} + 841 T^{4} )^{2} \))
$31$ (\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 - 4 T - 43 T^{2} + 12 T^{3} + 2024 T^{4} + 372 T^{5} - 41323 T^{6} - 119164 T^{7} + 923521 T^{8} \))(\( 1 - 4 T - 43 T^{2} + 12 T^{3} + 2024 T^{4} + 372 T^{5} - 41323 T^{6} - 119164 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))(\( ( 1 - 8 T + 37 T^{2} )^{2}( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} ) \))(\( ( 1 - 8 T + 37 T^{2} )^{2}( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} ) \))
$41$ (\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 28 T^{2} - 246 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 6 T + 28 T^{2} + 246 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{4} \))(\( ( 1 - 5 T + 43 T^{2} )^{4} \))
$47$ (\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 + 6 T - 11 T^{2} + 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 4 T^{2} + 324 T^{3} - 2301 T^{4} + 15228 T^{5} - 8836 T^{6} - 622938 T^{7} + 4879681 T^{8} \))(\( 1 + 6 T - 4 T^{2} - 324 T^{3} - 2301 T^{4} - 15228 T^{5} - 8836 T^{6} + 622938 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( ( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} )^{2} \))(\( ( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} )^{2} \))
$59$ (\( 1 - 3 T - 50 T^{2} - 177 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4} \))(\( 1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4} \))(\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 + 22 T + 252 T^{2} + 2508 T^{3} + 21787 T^{4} + 147972 T^{5} + 877212 T^{6} + 4518338 T^{7} + 12117361 T^{8} \))(\( 1 - 22 T + 252 T^{2} - 2508 T^{3} + 21787 T^{4} - 147972 T^{5} + 877212 T^{6} - 4518338 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( 1 - 61 T^{2} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + 20 T + 185 T^{2} + 1860 T^{3} + 18104 T^{4} + 113460 T^{5} + 688385 T^{6} + 4539620 T^{7} + 13845841 T^{8} \))(\( 1 + 20 T + 185 T^{2} + 1860 T^{3} + 18104 T^{4} + 113460 T^{5} + 688385 T^{6} + 4539620 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \))(\( 1 - T - 66 T^{2} - 67 T^{3} + 4489 T^{4} \))(\( 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} \))(\( 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} \))(\( 1 - T - 66 T^{2} - 67 T^{3} + 4489 T^{4} \))(\( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \))(\( 1 + 6 T - 79 T^{2} - 114 T^{3} + 6324 T^{4} - 7638 T^{5} - 354631 T^{6} + 1804578 T^{7} + 20151121 T^{8} \))(\( 1 + 6 T - 79 T^{2} - 114 T^{3} + 6324 T^{4} - 7638 T^{5} - 354631 T^{6} + 1804578 T^{7} + 20151121 T^{8} \))
$71$ (\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 26 T + 304 T^{2} - 1846 T^{3} + 5041 T^{4} )^{2} \))(\( ( 1 + 26 T + 304 T^{2} + 1846 T^{3} + 5041 T^{4} )^{2} \))
$73$ (\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( 1 + 13 T + 96 T^{2} + 949 T^{3} + 5329 T^{4} \))(\( 1 + 13 T + 96 T^{2} + 949 T^{3} + 5329 T^{4} \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 - 34 T^{2} - 4173 T^{4} - 181186 T^{6} + 28398241 T^{8} \))(\( 1 - 34 T^{2} - 4173 T^{4} - 181186 T^{6} + 28398241 T^{8} \))
$79$ (\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( 1 - 3 T - 70 T^{2} - 237 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - 4 T + 29 T^{2} + 684 T^{3} - 7336 T^{4} + 54036 T^{5} + 180989 T^{6} - 1972156 T^{7} + 38950081 T^{8} \))(\( 1 - 4 T + 29 T^{2} + 684 T^{3} - 7336 T^{4} + 54036 T^{5} + 180989 T^{6} - 1972156 T^{7} + 38950081 T^{8} \))
$83$ (\( ( 1 + 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 7 T + 83 T^{2} )^{2} \))(\( ( 1 - 7 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))(\( ( 1 + 16 T + 202 T^{2} + 1328 T^{3} + 6889 T^{4} )^{2} \))(\( ( 1 - 16 T + 202 T^{2} - 1328 T^{3} + 6889 T^{4} )^{2} \))
$89$ (\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 88 T^{2} + 324 T^{3} + 4251 T^{4} + 28836 T^{5} - 697048 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( 1 + 6 T - 88 T^{2} - 324 T^{3} + 4251 T^{4} - 28836 T^{5} - 697048 T^{6} + 4229814 T^{7} + 62742241 T^{8} \))
$97$ (\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 17 T + 97 T^{2} )^{2} \))(\( ( 1 + 5 T + 97 T^{2} )^{2} \))(\( ( 1 + 5 T + 97 T^{2} )^{2} \))(\( ( 1 - 17 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 6 T + 91 T^{2} - 582 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 6 T + 91 T^{2} - 582 T^{3} + 9409 T^{4} )^{2} \))
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